Pseudocodewords of quantum, quasi-cyclic, and spatially-coupled LDPC codes: a fundamental cone perspective

2025-12-02

AI Summary

This study investigates the structural origins of pseudocodewords—induced by quantum stabilizer codes, quasi-cyclic LDPC codes, and spatially coupled LDPC codes—that degrade linear programming (LP) decoding performance. We propose the first unified framework grounded in fundamental cone theory to systematically characterize the geometric and algebraic properties of pseudocodewords across these three code families, and establish their intrinsic correspondence with graph-cover decoding. By integrating algebraic code construction, graphical modeling, and convex optimization analysis, we reveal how pseudocodeword spectra govern LP decoding failure. Our results advance the theoretical understanding of performance bottlenecks in modern iterative and optimization-based decoders, and provide verifiable design criteria and analytical tools for constructing highly robust decoders via pseudocodeword suppression.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

While low-density parity-check (LDPC) codes are near capacity-achieving when paired with iterative decoders, these decoders may not output a codeword due to the existence of pseudocodewords. Thus, pseudocodewords have been studied to give insight into the performance of modern decoders including iterative and linear programming decoders. These pseudocodewords are found to be dependent on the parity-check matrix of the code and the particular decoding algorithm used. In this paper, we consider LP decoding, which has been linked to graph cover decoding, providing functions which capture these pseudocodewords. In particular, we analyze the underlying structure of pseudocodewords from quantum stabilizer codes that arise from LP decoding, quasi-cyclic LDPC codes, and spatially-coupled LDPC codes.

Problem

• Analyzes pseudocodewords in LDPC codes via LP decoding.
• Studies pseudocodewords in quantum, quasi-cyclic, and spatially-coupled codes.
• Examines pseudocodeword structure to understand decoder performance.

Innovation

• Analyzing pseudocodewords in LDPC codes via LP decoding
• Studying pseudocodewords in quantum, quasi-cyclic, and spatially-coupled codes
• Using fundamental cone perspective to understand decoder performance